Fixed point method for nonlinear systems - complex roots

[RicardoMP]

Homework Statement

I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point method with the iterative function G (that I've also uploaded). I must also justify the convergence or divergence of its iterations.

Homework Equations

The F and G functions are in the .png files that I've uploaded, where x=(x1,x2).

The Attempt at a Solution

Before I tried to verify the convergence criteria for the fixed point method, I tried to find the roots and proceeded to rewrite G in the form x=G(x). And so I arrived to the expressions: $x_1=\pm \sqrt{-x_2^2}$ and $x_2=\pm \sqrt[4]{\frac{1-x_1^2}{4}}$.
Obviously faced with complex roots, before applying the fixed point method, I went on and tried to verify the convergence of its iterations, by studying the max norm of the Jacobian matrix of the rewritten G function. In the end, I got the matrix that I've uploaded (systemJG.png). Given a certain real interval to $x_1$ and $x_2$, I have to compare the absolute value of a complex number and a real number. Anyway, I think I'm not going in the right path and I'm overlooking something and complicating the whole problem, starting with the fact I've only considered the positive roots of the $x_1$ and $x_2$ in the matrix... Any tips in how to tackle this problem or any similar example I can study?

 

[mighty2000]

Hello,

Thank you for sharing your approach and progress on this problem. It seems like you have made some good progress so far. However, I would suggest taking a step back and thinking about the overall goal of the problem. You have been asked to approximate the roots of the system of equations using the fixed point method and justify the convergence or divergence of its iterations.

One approach to this problem would be to first use the fixed point method to approximate the roots of the system. This involves iterating the function G until we reach a point where G(x) = x, which would indicate that x is a root of F. This can be done by starting with an initial guess for x and then repeatedly applying G to it. It is important to choose an appropriate initial guess to ensure convergence of the iterations.

Once you have approximated the roots, you can then analyze the convergence or divergence of the iterations by looking at the behavior of G. In this case, you can use the Jacobian matrix of G to determine the stability of the fixed points. If the absolute value of the eigenvalues of the Jacobian matrix is less than 1, then the iterations will converge to the fixed point. If the absolute value is greater than 1, then the iterations will diverge. This can also be visualized by plotting the function G and seeing where it intersects with the line y=x.

I hope this helps and provides some direction for your approach. Good luck!